Is a model nested with itself before collapsing categorical variables?

[mynameisfunk]

If I have a model with a categorical variable X1=0,1,2,3 and a continuous variable X2 and I have a regression model that includes an interaction between X1 and X2, then I decide I want to collapse X1 into an indicator variable where X1_new=0 if X=0 and X1_new=1 if X>=1, is my new model with an interaction term between X1_new and X2 nested with the original model?

 

[Stephen Tashi]

You should quote the definitions for "interaction term" and "nested model" that you are using.

You might mean that you are dealing with a quadratic model of the form Model_A: Z = A X1 + B X2 + C X1 X2 + D with variables X1, X and another model of the form Model_B: Z = P Y1 + Q X2 + R Y1 X2 + S with variables Y1 and X2 where X2 is the same variable as in Model_A and Y1 is the function of X2 given by Y1 = 0 if X1 = 0 and Y1 = 1 otherwise.

The definition of "Model_B is nested within Model_A" might mean that there exists a way to write Model_A and model_B as quadratic models using the same set of variables such that Model_B expresses Z as a proper subset of the "terms" in Model_A. But does this also mean that a term (such as Q X2) that appears in Model_B has the same (perhaps unknown) constant coefficient as the corresponding term in Model_B (i.e. must P = Q in the previous example?).